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IB DP Physics · Theme B The particulate nature of matter · B.1 · ~8 min read

B.1 Thermal energy transfers

The particle model, density, Kelvin temperature and internal energy, specific heat capacity and latent heat, conduction, convection and radiation, the Stefan–Boltzmann law, luminosity, apparent brightness and Wien's law.

Particle modelInternal energySpecific heat & latent heatConductionConvectionBlack-body radiationLuminosity & brightnessWien's law
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • Matter is made of particles in constant random motion. Temperature in kelvin measures the average random kinetic energy of the particles: Ek = 32kBT.
  • Internal energy = total random kinetic energy + intermolecular potential energy. Heating raises the temperature (Q = mcΔT) or changes the phase at constant temperature (Q = mL).
  • Thermal energy flows from hot to cold by conduction, convection and radiation. Conduction rate: ΔQ/Δt = kAΔT/Δx.
  • Hot bodies radiate: a black body's power is L = σAT4, its peak wavelength obeys λmaxT = 2.9 × 10−3 m K, and the brightness seen from a distance d is b = L/4πd2.
  • B.1 is the same at SL and HL.

By the end you can

  • Describe solids, liquids and gases with the particle model and use ρ = m/V.
  • Convert between Celsius and kelvin and relate temperature to average kinetic energy.
  • Solve heating and phase-change problems with specific heat capacity and specific latent heat.
  • Explain conduction, convection and radiation, and use the conduction equation.
  • Use the Stefan–Boltzmann law, apparent brightness and Wien's displacement law.

1The particle model

solidfixed positions, vibrateliquidclose, slide pastgasfar apart, fast, random
Same particles, different arrangement and energy. Changing phase changes the spacing, not the particles.
ρ = mV DATA BOOKLET
density in kg m−3
  • Kelvin and Celsius: T/K = θ/°C + 273. A change of 1 K equals a change of 1 °C. Absolute zero (0 K) is the temperature of minimum internal energy.
Ek = 32kBT DATA BOOKLET
average translational kinetic energy of a particle; kB = 1.38 × 10−23 J K−1

Internal energy — the total of the random kinetic energies and the intermolecular potential energies of all the particles in a body.

Worked example

Find the average kinetic energy of a gas molecule at 27 °C.

  1. T = 27 + 273 = 300 K A1
  2. Ek = 1.5 × 1.38 × 10−23 × 300 M1
  3. = 6.2 × 10−21 J A1
Common mistakeUsing °C in Ek = 32kBT, PV = nRT or σT4. Every thermal equation with T in it needs kelvin.

2Heating and phase change

Q = mcΔT DATA BOOKLET
c = specific heat capacity, J kg−1 K−1
Q = mL DATA BOOKLET
L = specific latent heat (fusion or vaporisation), J kg−1
timeθ / °C-20020406080100120melting: Q = mLfboiling: Q = mLvQ = mcΔTicesteamFlat parts: energy goes into breaking bonds, so temperature is constant.
Heating ice at a steady rate. The temperature only rises when the phase is not changing.
  • During a phase change the energy supplied increases the potential energy of the particles (bonds are broken), so the average kinetic energy, and therefore the temperature, stays constant.
Worked example

A 2.0 kW kettle heats 1.5 kg of water from 20 °C to 100 °C (c = 4200 J kg−1 K−1). How long does it take if no energy is lost?

  1. Q = 1.5 × 4200 × 80 = 5.04 × 105 J M1A1
  2. t = Q/P = 5.04 × 105/2000 = 252 s (about 4 minutes) A1
Worked example

0.30 kg of water at 80 °C is mixed with 0.20 kg of water at 20 °C. Find the final temperature.

  1. energy lost by hot water = energy gained by cold water M1
  2. 0.30 × c × (80 − T) = 0.20 × c × (T − 20) A1
  3. 24 − 0.30T = 0.20T − 4 ⇒ T = 56 °C A1
Worked example

How much energy melts 0.20 kg of ice at 0 °C and then warms it to 10 °C? (Lf = 3.34 × 105 J kg−1)

  1. melting: 0.20 × 3.34 × 105 = 66 800 J M1
  2. warming: 0.20 × 4200 × 10 = 8 400 J M1
  3. total = 7.5 × 104 J A1
Exam tip"Explain why the temperature stays constant during melting" needs two ideas: energy goes into potential energy (breaking bonds) and the kinetic energy, which temperature measures, does not change.

3Conduction, convection and radiation

hotT₁coldT₂energy flow ΔQ/Δtthickness Δx, area A (end face)ΔQ/Δt = kA ΔT/Δx: better conductor (large k), bigger area, steeper gradient ⇒ faster
Fast-vibrating particles at the hot end pass energy to slower neighbours. In metals, free electrons carry most of it.
ΔQΔt = kAΔTΔx DATA BOOKLET
k = thermal conductivity, W m−1 K−1
Worked example

A window has area 1.5 m2 and glass 4.0 mm thick (k = 0.80 W m−1 K−1). Inside is 20 °C, outside 5 °C. Find the rate of energy loss.

  1. ΔT/Δx = 15/0.0040 = 3750 K m−1 M1
  2. ΔQ/Δt = 0.80 × 1.5 × 3750 M1
  3. = 4.5 kW (double glazing traps air, k ≈ 0.025, to cut this) A1
  • Convection happens in fluids: the heated fluid expands, its density falls, and it rises; cooler, denser fluid sinks to replace it. No convection in solids or in a vacuum.
  • Radiation is electromagnetic (mostly infrared at everyday temperatures). It needs no medium, which is how the Sun's energy reaches the Earth.

Black-body radiation

Black body — an ideal object that absorbs all radiation falling on it and emits the maximum possible radiation for its temperature.

L = σAT4 DATA BOOKLET
luminosity (total power emitted); σ = 5.67 × 10−8 W m−2 K−4
λmaxT = 2.9 × 10−3 m K DATA BOOKLET
Wien's displacement law
λ / μmintensity0.511.522.5visible6000 K (blue), 5000 K, 4000 Khotter: peak moves to shorter λand total power rises as T⁴λmaxT = 2.9 × 10⁻³ m K
Hotter bodies emit more at every wavelength, and their peak moves towards the blue.
Worked example

The Sun's surface is at 5800 K and its radius is 6.96 × 108 m. Find its luminosity and peak wavelength.

  1. A = 4πR2 = 6.09 × 1018 m2 M1
  2. L = 5.67 × 10−8 × 6.09 × 1018 × 58004 = 3.9 × 1026 W A1
  3. λmax = 2.9 × 10−3/5800 = 5.0 × 10−7 m (500 nm, visible) A1

Luminosity and apparent brightness

star, Ld2d3db = L/(4πd²)double d:b falls to ¼triple d: 1/9
The same power spreads over a sphere of area 4πd2, so brightness falls with the square of distance.
b = L4πd2 DATA BOOKLET
apparent brightness, W m−2
Worked example

Find the Sun's apparent brightness at the Earth (d = 1.5 × 1011 m).

  1. b = 3.9 × 1026/(4π × (1.5 × 1011)2) M1
  2. ≈ 1.4 × 103 W m−2: the solar constant used in B.2 A1
Common mistakeUsing the radius of the star in b = L/4πd2. d is the distance from the star to the observer; R appears only in L = σ(4πR2)T4.
Linking questionThe same black-body ideas power the greenhouse model in B.2 and the stellar HR diagram in E.5, where two stars of the same colour but different brightness must have different radii.
Idea for your IA / explorationMeasure the specific heat capacity of a metal block with an immersion heater, joulemeter and thermometer. Plot temperature against energy supplied, use the gradient, and discuss heat losses as a systematic error.

Test yourself — cover the answers first

Quick check

  1. Convert −40 °C to kelvin.
  2. What happens to the temperature of water while it boils?
  3. Which transfer mechanism works in a vacuum?
  4. Two stars have the same temperature; one has twice the radius. Ratio of luminosities?
  5. A star's peak wavelength halves. What happens to its surface temperature?
  6. What is the effect on apparent brightness of tripling the distance?

Exam-style

  1. A 0.50 kg copper block (c = 385 J kg−1 K−1) at 200 °C is dropped into 0.40 kg of water at 15 °C. Find the final temperature (ignore losses).
  2. A star has luminosity 2.0 × 1028 W and surface temperature 12 000 K. Find its radius.
Show answers
  1. 233 K
  2. It stays constant (100 °C at standard pressure).
  3. radiation
  4. 4 (L ∝ R2 at fixed T)
  5. It doubles.
  6. falls to 1/9
  7. 0.50 × 385 × (200 − T) = 0.40 × 4200 × (T − 15)
    38 500 − 192.5T = 1680T − 25 200
    T = 63 700/1872.5 = 34 °C
  8. R2 = L/(4πσT4) = 2.0 × 1028/(4π × 5.67 × 10−8 × 12 0004)
    = 1.35 × 1018 m2
    R = 1.2 × 109 m (about 1.7 solar radii)
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.